The concept of decompositions that we explore here is related to systems that can be defined by their given input and produced output.
We model these as an input domain \(\mathcal {I}\)  and an output domain \(\mathcal {O}\) , not necessarily finite. Our description of a system is
a specification that associates inputs to outputs, and is modeled as a relation \(R\subseteq \mathcal {I}\times \mathcal {O}\)  [1]}, [2]}.
In addition we assume a constraint in form of a domain with a specific size that directs the decomposition to be more concise
and is given as an intermediate domain \(\mathcal {B}\) .
The objective is to decompose \(R\)  into relations \(R_1\subseteq \mathcal {I}\times \mathcal {B}\) , and \(R_2\subseteq \mathcal {B}\times \mathcal {O}\)  such that the composition \(R_1\circ R_2\)  has either of the following properties: (i) no input-output association is added or lost - this problem is called the Total Decomposition Problem (TDP), or (ii) a more relaxed version, called the Partial Decomposition Problem (PDP) in which no input-output association is added, but we are allowed to lose some of the output as long as each input can be resolved.
